CSCE475/875 Multiagent Systems

Handout 14: The Coalitional Core and The Shapley Value

November 3, 2011

(Based on Shoham and Leyton-Brown 2011)

 

The Core

First, let denote a mapping from a coalitional game (that is, a set of agents  and a value function ) to a vector of  real values, and let  denote the th such real value. Denote such a vector of  real values as . Each  denotes the share of the grand coalition’s payoff that agent  receives. When the coalitional game  is understood from context, we write  as a shorthand for .

 

The Shapley value defined a fair way of dividing the grand coalition’s payment among its members. However, this analysis ignored questions of stability. We can also ask:

 

Would the agents be willing to form the grand coalition given the way it will divide payments, or would some of them prefer to form smaller coalitions?

 

Unfortunately, sometimes smaller coalitions can be more attractive for subsets of the agents, even if they lead to lower value overall. Considering the majority voting example in Handout 13, while  does not have a unilateral motivation to vote for a different split,  and  have incentive to defect and divide the  million between themselves (e.g., dividing it

 

This leads to the question of what payment divisions would make the agents want to form the grand coalition. The answer is that they would want to do so if and only if the payment profile is drawn from a set called the core, defined as follows.

 

Definition 12.2.9 (Core) A payoff vector  is in the core of a coalitional game  if and only if

 

Thus, a payoff is in the core if and only if no sub-coalition has an incentive to break away from the grand coalition and share the payoff it is able to obtain independently.

 

That is, it requires that the sum of payoffs to any group of agents must be at least as large as the amount that these agents could share among themselves if they formed a coalition on their own. Notice that Definition 12.2.9 implies that payoff vectors in the core must always be imputations: that is, they must always be strictly budget balanced and individually rational.

 

 

 

 

Two Questions about the Core

As a notion of stability for coalitional games, the core is appealing. However, there are two important questions:  1. Is the core always nonempty?  2. Is the core always unique?  Unfortunately, the answer to both questions is no.

 

Example. Let us consider again the Parliament example with the four political parties. The set of minimal coalitions that meet the required 51 votes is  and . We can see that if the sum of the payoffs to parties , and  is less than $100 million, then this set of agents has incentive to deviate. On the other hand, if  and  get the entire payoff of $100 million, then  will receive $0 and will have incentive to form a coalition with whichever of , and  obtained the smallest payoff. Thus, the core is empty for this game.

 

These examples call into question the universality of the core as a solution concept for coalitional games. We already saw in the context of noncooperative game theory that solution concepts—notably, the Nash equilibrium—do not yield unique solutions in general.  Here we are in an arguably worse situation, in that the solution concept may yield no solution at all.

 

Can we characterize when a coalitional game has a nonempty core? Yes!  To do so, we first need to define a concept known as balancedness.

 

Definition 12.2.10 (Balanced weights) A set of nonnegative weights (over ), , is balanced if

 

Intuitively, the weights on the coalitions involving any given agent  can be interpreted as the conditional probabilities that these coalitions will form, given that  will belong to a coalition.

 

Theorem 12.2.11 (Bondereva–Shapley) A coalitional game has a nonempty core if and only if for all balanced sets of weights                                                                      

 

Theorem 12.2.12 Every constant-sum game that is not additive has an empty core.

 

We say that a player is a veto player if .

 

Theorem 12.2.13 In a simple game the core is empty iff there is no veto player. If there are veto players, the core consists of all payoff vectors in which the nonveto players get zero.

 

Theorem 12.2.14 Every convex game has a nonempty core.

 

The Core and the Shapley Value

If the core is not empty, is the Shapley value guaranteed to lie in the core? The answer in general is no, but the following theorem gives us a sufficient condition for this property to hold.

 

Theorem 12.2.15  In every convex game, the Shapley value is in the core.